01 - Preheat oven to 400°F.
02 - In a large Dutch oven or heavy pot, melt butter with olive oil over medium heat. Add onion, carrots, and celery. Cook 5–7 minutes until softened. Stir in garlic and cook 1 minute.
03 - Sprinkle flour over vegetables; stir and cook for 2 minutes.
04 - Gradually whisk in chicken broth, stirring to avoid lumps. Add milk and bring to a gentle simmer.
05 - Add shredded chicken, peas, corn, thyme, parsley, pepper, and salt. Simmer 10–15 minutes, stirring occasionally, until thickened.
06 - In a bowl, whisk together flour, baking powder, baking soda, and salt. Cut in cold butter with a pastry blender or fingertips until mixture resembles coarse crumbs. Stir in buttermilk and parsley just until dough comes together.
07 - Turn dough onto a floured surface. Pat to 3/4-inch thickness. Cut into 2-inch rounds or squares.
08 - Ladle soup into a large oven-safe baking dish or individual ramekins. Arrange biscuits on top, leaving a little space between each.
09 - Brush biscuit tops with additional buttermilk. Bake 20–25 minutes, until biscuits are golden and soup is bubbling.
10 - Cool slightly before serving.